Answer :
To set up an integral for the area of the shaded region, we need to consider the shape of the region and decide on an appropriate coordinate system.
Let's assume that the shaded region is bounded by the curve y=f(x) and the x-axis, and that it lies between the vertical lines x=a and x=b.
Then the area of the shaded region can be expressed as the following definite integral:
Area = ∫[a,b] f(x) dx
To evaluate this integral, we can use the Fundamental Theorem of Calculus:
Area = F(b) - F(a)
where F(x) is an antiderivative of f(x).
To find the area of the shaded region, we would need to find an antiderivative of f(x) and then evaluate the integral by substituting the appropriate values for a, b, and F(x)
For example, if f(x) = x^2 and a = 0 and b = 2, then the area of the shaded region would be:
Area = ∫[0,2] x² dx
= F(2) - F(0)
[tex]=\frac {(2^3)}3 - \frac{(0^3)}3\\ =\frac{8}{3}[/tex]
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